TITLE: Difference of powers inequality
QUESTION [6 upvotes]: Is there any inequality involving difference of powers?  I want to estimate say $x^p - y^p$, or $|x^p -y^p|$ hopefully in terms of $|x-y|^p$
Edit : $0 < p <\infty$, and I'm sure that there will be cases depending on $p$

REPLY [2 votes]: In fact, we have the following inequalities:

If $x,y>0$, then $|x-y|^p\le p|x-y|(x^{p-1}+y^{p-1}), \text{ for }p\ge 1$,
If $x\ge y>0$, then $|x-y|^p\le \frac{p}{2}x^{p-2}|x-y|(x^2+y^2), \text{ for }p\ge 2$.
(Mazur Inequality) For any real $x,y$, then $2^{1-p}|x-y|^p\le\big|x|x|^{p-1}-y|y|^{p-1}\big|\le|x-y|(|x|^{p-1}+|y|^{p-1}), \text{ for }p\ge 1$.